128 Mr WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE, 



27. The application of the formulae in these tables, and many 

 problems in Astronomy, require the solution of this problem in the 

 Calculus of Sines: 



Given the s?m or differetice of two angles, and the ratio of their 

 sines, to find the angles. 



This problem is identical with a case of Plane Trigonometry ; but it 

 may be elegantly resolved without any reference to Trigonometry, by 

 means of subsidiary angles. 



Let <p and v|/ be the two angles : there are given (p + ^\^, or else 

 d) — xi/, and -. — —, to determine d) and \fr. 



First solution. Let e be an angle, such that 



sin \//_ 



sm e = 



sin (p ' 



tan \{<p + ^) 



Second. Take e such that cos e 



sin x|/ 

 sin (J) 



then tan t (</'-'/-) ^ ^an^ 



,2 1 



(!)• 



(2). 



Third. Find e so that tan e = 



sin >// 

 sin <}) 



then f"fg-f = tan(45°-.) 

 tan ^{(p-i') 



Fourth. Take, tan e = V^^^; 



^ sm <p 



then, f^!-^^, = cos 2e ( 

 . tani((^->/.) j 



(3). 



(4). 



